Possible but rare. After one dose of chickenpox vaccine, most children will have enough immunity that they won't get chickenpox even if they are exposed. After two doses, at least 95% of kids have good immunity against the disease. So, there is a slight chance for some kids to get chickenpox even with vaccination. In those cases, the disease is usually pretty mild.

Chickenpox vaccine. When the chickenpox vaccine came out, the belief was that the recipient will be immune to chickenpox. However, as time went by, came reports that some children who had the vaccine still caught the disease. Because of this, a second dose of the vaccine (booster) was recommended to boost the immunity with the aim of preventing future reinfection. The same situation happened with the measles vaccine.